where \(g({\bf {n}},{\bf {m}},{\bf {t}})\ge 0\) .
Assume further that there exists
a homeomorphism \({\bf {n}}\rightarrow \tilde{\bf {n}}\)  on \(\mathcal {K}\)  such that
\(P_{\tilde{\bf {n}}}=\overline{P_{\bf {n}}}\)  for every \({\bf {n}}\in \mathcal {K}\) .
In this case \(\mathcal {K}\)  with the convolution of two point measures defined by
\(\varepsilon _{\bf {n}}\ast \varepsilon _{\bf {m}}(\varepsilon _{\bf {t}}):=p({\bf {n}},{\bf {m}})(\varepsilon _{\bf {t}}):=g ( {\bf {n}},{\bf {m}}, {\bf {t}})\)  is a
hypergroup which is called a polynomial hypergroup in \(d\) 
variables. The hypergroup \(\mathcal {K}\) 
is obviously commutative and the identity element \(e\)  is the constant
polynomial \(P_0 \equiv 1.\)  The character space \(\widehat{\mathcal {K}}\)  can
be identified with the set
\( \lbrace  {\bf {x}}\in {C}^d: | \alpha _{\bf {x}}({\bf {n}})|\le 1, \alpha _{\bf {x}}(\tilde{\bf {n}}) =\overline{\alpha _{\bf {x}}({\bf {n}})} \hspace{5.69046pt} \forall {\bf {n}}\in \mathcal {K} \rbrace \) ,
where
\(\alpha _{\bf {x}}({\bf {n}}):=P_{{\bf {n}}}({\bf {x}})\) 
for \({\bf {x}}\in {C}^d\)  and \({\bf {n}}\in \mathcal {K}\) .
For more
on polynomial
hypergroups in several variables we refer
the reader to e.g. [1]}, [2]}, [3]}.
